
where we used Ito calculus to expand the cost-to-go function, \(J({\bf x}_{t+\delta t},t+\delta t)\) ; see e.g. ref. [1]}. By reordering the terms in eq. REF , we arrive at the Hamilton-Jacobi-Bellman (HJB) equation,
\(\nonumber -\frac{\partial }{\partial t} J({\bf x},t) &=&\min _{\bf u}\left[ \Omega ({\bf x}_t ,{\bf u}_t)+ (A({\bf x}_t)+{\bf u})^\top \cdot \nabla J + \frac{1}{2}{\text{\bf Tr}}K\nabla _{{\bf x}{\bf x}} J\right]\\&=& \min _{\bf u}\left[ \frac{1}{2} {\bf u}^\top B {\bf u}+{\bf u}^\top \cdot \nabla J \right]+V({\bf x})+ A({\bf x}_t)^\top \cdot \nabla J + \frac{1}{2}{\text{\bf Tr}}K \nabla _{{\bf x}{\bf x}} J \) 
